Gaussian Integral

The Gaussian integral, otherwise known as the Euler-Poisson or probability integral, is the definite integral of the Gaussian function, $e^{x^2}$, over $(-\infty,\infty)$. This particular definite integral arises often when performing statistical calculations and when normalizing quantum mechanic wave functions. Here, the value of the Gaussian integral is derived through double integration in polar coordinates, namely shell integration.

Consider the Gaussian integral, $\int_{-\infty}^{\infty} e^{-x^2} dx$. Using heuristic methods, this integral is difficult if not impossible to solve. However, notice that taking the square of the integral allows us to write this equation:

\begin{equation}
\left(\int_{-\infty}^{\infty} e^{-x^2} dx\right)^2 = \int_{-\infty}^{\infty} e^{-x^2} dx \int_{-\infty}^{\infty} e^{-x^2} dx
\end{equation}

Since the integrals are definite with respect to $x$, $x$ can be replaced by any dummy variable of choice. This allows the substitution of $y$ in the place of $x$ in the second definite integral in the right side of the above equation.

\begin{equation}
\left(\int_{-\infty}^{\infty} e^{-x^2} dx\right)^2 = \int_{-\infty}^{\infty} e^{-x^2} dx \int_{-\infty}^{\infty} e^{-y^2} dy
\end{equation}
\begin{equation}
\left(\int_{-\infty}^{\infty} e^{-x^2} dx\right)^2 = \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} e^{-\left(x^2 + y^2\right)} dx dy
\end{equation}

The right side of the above equation is a double integral in Cartesian coordinates and can be equivalently expressed using shell integration (a double integration in polar coordinates). Let $r\cos\theta = x$ and $r\sin\theta = y$, making $r^2 = x^2 + y^2$. Writing the double integral of equation $(3)$ using shell inegration yields:

\begin{equation}
\left(\int_{-\infty}^{\infty} e^{-x^2} dx\right)^2 = \int_0^{2\pi} \int_{0}^{\infty} e^{-r^2} r dr d\theta
\end{equation}

Let $u=r^2$.

\begin{equation}
\left(\int_{-\infty}^{\infty} e^{-x^2} dx\right)^2 = \frac{1}{2} \int_0^{2\pi} \int_{0}^{\infty} e^{-u} du d\theta
\end{equation}
\begin{equation}
\left(\int_{-\infty}^{\infty} e^{-x^2} dx\right)^2 = \frac{1}{2} \int_0^{2\pi} d\theta
\end{equation}
\begin{equation}
\left(\int_{-\infty}^{\infty} e^{-x^2} dx\right)^2 = \pi
\end{equation}
\begin{equation}
\int_{-\infty}^{\infty} e^{-x^2} dx = \sqrt{\pi}
\end{equation}